


z ↔ z² + ci (a series of transformations)

by kusuriurikun



Series: Shibuya Continuum [1]
Category: Subarashiki Kono Sekai | The World Ends With You
Genre: #TWEWY13, #ThankYouTWEWY, Canonical Character Death, Character Study, During Canon, Gen, I let Sho say "fuck" so be forewarned, Math and Science Metaphors, Mathematics, POV Antagonist, Platonic Relationships, Post-Reapers' Game (TWEWY), Pre-Canon, TWEWY Anniversary 2020, is there a tag for "forbidden magicks via mathematics", let's just face it it's Sho's entire life and post-death "life"
Language: English
Status: Completed
Published: 2020-07-25
Updated: 2020-07-25
Packaged: 2021-03-04 23:41:32
Rating: Teen And Up Audiences
Warnings: Major Character Death, No Archive Warnings Apply
Chapters: 8
Words: 8,893
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/25434808
Author URL: https://archiveofourown.org/users/kusuriurikun/pseuds/kusuriurikun
Summary: A look at the complex function that is Sho Minamimoto, in a series of mathematical transformations and operations.A bit of an experimental work, so please forgive the fact this may in fact resemble one of the infamous "heaps" in practice.Posted for the thirteenth anniversary of The World Ends With You and in celebration of the upcoming anime adaptation.Rated Teen for adult language and (rather inherent to the process of becoming a Reaper) death.Due to some of the complexity of math (and esoteric) concepts played with, appendix and author's notes attached at end.
Series: Shibuya Continuum [1]
Series URL: https://archiveofourown.org/series/1902298
Comments: 8
Kudos: 18





	1. hyper0: SUCCESSOR

The kid was always a weird one in retrospect. Quiet, except when he wasn’t, and then the little brat wouldn’t shut up; where most kids his age were into dinosaurs, or Doraemon, orr the like...he’d be spinning wheels on the toy cars, or looking in books about numbers or other neat stuff...never did get along with the other kids.

At one point, when he was in elementary school...the counselor called his parents. Took them aside. Let them know how sometimes they’d find Sho doing...odd things, not _bad_ , just... _odd_. Staring at the room fan. Counting almost obsessively. Had Issues taking turns. Had Issues being touched sometimes. They had some worries on how this could impact him academically--had they thought about having him possibly evaluated, to see if he had some kind of issue?

The father (one Mr. Iwa Minamimoto, whom was notable primarily as a salaryman at a no-name black company) noted the only _issue_ he felt young Sho had was that he was _ornery_ , and certainly did not require _psychological evaluation_.

And that, as it was, was that.

* * *

At some point in his life, young Sho Minamimoto has one of those life-changing formative experiences that sets the function of one’s life on a particular asymptote.

Namely, a teacher (who sees his fixations) hands him a book about numbers, but not just the usual sort--things like _pi_ and _e_ and _n!_ that his dad’s calculator (that he was never, _ever_ allowed to touch) had on it, and even some stuff about a thing called a “fractal” which was basically a Thing made of it repeating itself in smaller bits down to forever. A simple book about how there’s math in literally everything, on some level or another.

The teacher had worried a bit--usually this is the sort of thing they’d not introduce until secondary school, really, year 3 was probably a bit young for this...but it’d keep the kid busy for a while, at any rate, and there was no harm in sparking his curiosity.

This...this brought a new world. A new series of things to Explore. And it was Sho’s first introduction to something he will see as pretty much THE one great axiom in his life:

_The world is made of numbers. Do the math, and you’ll arrive at your desired solution._


	2. hyper1: ADDITION

As time went on--as any function of time goes, really--things change, grow in complexity.

As his own timeline continues, Sho discovers even more deep beauty in mathematics. He learns at some point about the Pythagoreans, who built an entire philosophy around math being the rules of the world and a beauty in and of itself, its own chorus--he can appreciate this in his own way, and it is one of the things that brings him deep comfort in life.

He discovers Youtube streamers of people with similar fascinations with numbers, with math, to himself. Plumbing not only _π_ (which was always one of his favourite numbers) but the natural logarithmic base _e_ and the imaginary base _i_. The day he discovers Euler’s Identity in some musty book in the library is an epiphany to him-- _e iπ+1=0 _ and the realization that the three constants are intimately related in this way is to him a confirmation of that sense he has that the world IS made of numbers, and if he could just figure out the right equation, all would fall in place.

Just as beautifully as _e_ taken to the power of pi taken to the imaginary root, plus the multiplicative identity, equals zero.

He finds out about other things. Fractals also hold a special love, and chaos theory in general. Like many, he develops a fascination with the Mandelbrot set; the day he discovers that pi is actually hidden within, and that the Fibonacci sequence shows up...is again comfort, again a revelation that there is truth hidden in everything.

And there was so, so very much of it...the idea of hypercubes, which--in our space, anyways--look to be ever turning themselves in and out, a limitation of projecting a 4d surface in 3d--like the shadow of a sphere on a flat sheet. The whole idea you COULD essentially “cube a cube”. Things like tetration (which went beyond exponents, making power towers), and Graham’s number--which had to use a special notation for “power towers” in a ludicrous power tower in and of itself. (Sho actually finds the concept of a number requiring Knuth’s up-arrow notation extended to _64 layers_ to express it a bit of a grand joke, rather as he finds the infamous Hungarian inflation notes that included an actual one-sextillion-pengo bill.)

Math is always a comfort, and a source of truth and beauty, and is always the special interest that tends to draw Sho back in--and occasionally expands his own horizons, like his discovery of metal music via someone at the AMX suggesting albums the weird math kid would like, starting from when he picked up a Voivod album just from a mention of hypercubes. 

Prog metal would turn out to be an interest, too. The changes in signatures, the occasional chaotic shifts...it has its own mathematical beauty, too. (And we won’t even mention the day Sho found out about _The Holy Gift_ arrangement of Tool’s _Lateralus_ \--that, too, was an epiphany.)

Very late in his life, he discovers math zines. He’s one of the few people who hits Comiket _not_ for the tables of anime doujinshi but the booth for The Ankokudan.

Around this time, he also discovers that there are physicists who see the entire universe itself as a grand mathematical function--all constants, particles, quantum physics itself, being part of a grand and elegant computation.

Sho sees a great amount of elegance and truth in the mathematical universe hypothesis, even if he doesn’t quite (yet) understand all the equations involved.

* * *

As the asymptote of his life in terms of at least his intellectual curiosity rises, Sho finds his relationships with his family and with people in general taking a negative vector.

He learns to shut up around people in general, even with the math stuff. (Which, to him, is a goddamn tragedy. Sho REALLY wishes people could see what he sees in things like the simple relationship between the sine, cosine, and tangent and the balance of a right triangle.) Learns all too early that there are only a few people who see things in the way he does.

Even in those points he tries to reach out it blows up messily in his face more often than not. (The one guy who went “I prefer my music to be more than about _numbers_ ” when Sho was mentioning some math-rock band. The kids who pretty much STILL thought of him as the Weird Fucking Math Guy with all the goddamn triangles on his folder. The _teachers_ , half the time.) 

His old man--at least when his old man wasn’t at the office till 11pm working himself to an early grave by karoshi--is constantly on his case on _you’re so damn smart Sho, why do you have to be such a brat_ and _Goddamnit, Sho, I know you can apply yourself_ and _For the love of the kami, Sho, can you shut up about pi and fractals for_ **_five minutes_** _, why can’t you be_ **_normal_** _?_ and _Sho, you are never going to be anything but a NEET if you don’t get off the goddamn Internet and out of the books for five minutes, your goddamn fractals aren’t going to get you a job_. 

School...is _so zetta dull_. Courseloads geared towards getting kids on a college or vocational track, not exactly designed to encourage curiosity, the teachers pretty much gearing the course to the zeroes in class who are going to be lucky to manage a construction site or a 7-Eleven, and--especially starting in high school--Sho Minamimoto is, more often than not, bored out of his fucking _mind_. Increasingly, he fucks off and hits a library, or online, where he can learn far more than sitting in a classroom.

* * *

That said...despite the world, or at least human society, being garbage...there are still some bright spots. 

There’s the CAT mural--Sho does appreciate a lot of the themes there, and how CAT arts will just pop out of NOWHERE, and how pretty much nobody actually knows who the hell CAT is (is CAT a single artist? An artists’ circle? Something else?). There’s also just something... _comforting_ about it too--a bit of untamed, chaotic, spontaneous beauty in a plasticine and concrete cityscape that otherwise is the equivalent of a gilded garbage can.

There’s that blade of something distinctly weedy-yet-flowerish that’s coming up between the cracks in the pavement not too far from that CAT mural--the fact that such a little fractal can survive even here does give Sho some hope for himself.

There’s Doi’s ramen shop, where he can get a bowl and just get the hell away for an hour or so. (Though he does wonder what happened to the ram-crotch guy who used to work there as cook staff. Haven’t seen him in a while. Did he start his own place somewhere else?)

* * *

The last day of Sho Minamimoto’s mortal life pretty much begins as it tends to do--his dad yelling at him, proceeding to Fuck Off for the day, coming back home…

And for once his dad isn’t working himself into a heart attack at the age of 45 but is working himself into a froth at _home_ , and proceeds to lay into Sho on how he’s eighteen, and he doesn’t have to technically SUPPORT him, and if he’s not going to actually put in effort he needs to get a job or get out.

Sho tells his dad he’s not going to listen to this garbage, and stomps off. Figures it’s all garbage. He’ll head to Doi’s, and he’ll factor out from there what the fuck he’s gonna do…

Passes by the CAT mural. Stops for a bit. Sighs. Jesus, he wishes he could be like CAT sometimes. (Whoever CAT is, or are, that is.) Gets pissed seeing someone’s actually tried to deface the mural--the logo’s been freshly repainted. (Shame. He missed the cross and pentacle design. THAT...that was _zetta metal_ , probably why it pissed someone off enough it got defaced and CAT had to repaint.)

Sees that the little living fractal growing in the cracks in the pavement apparently got trampled between the last time he saw it and today. Yet more proof this world is garbage, in Sho’s opinion. Bastards wouldn’t know _beauty_ if it hit them.

Ponders his options. Maybe he could work for a math doujinshi circle...but yeah, he knows he can’t stay with his old man. He’s come to blows more than once. Pretty much fully expects when he comes back he’s gonna find all his shit in the little garbage piles on the street where he sometimes goes dumpster diving…

Doesn’t even bother looking at the crosswalk signal. He just wants to get fucking somewhere other than HERE. This place, this situation, is going to drive him zetta bugfuck nuts if he doesn’t get out…

And it is ironic that--in this cloud of frustration and rage--Sho Minamimoto does not notice the garbage truck hurtling towards him at approximately 20 m/s, ceasing _that_ particular iteration, and firmly locking him in the bounding box known as the UnderGround.


	3. hyper2: MULTIPLICATION

The last memory Sho has of being alive is of JUST having been hit and in a sea of pain, and he certainly does not remember in his first moments of his death--his unlife, his _existence in the complex plane_ \--how the hell he ended up in the Scramble Crossing to begin with.

Especially as he _wasn’t_ there when the wall of pain hit him and before he woke up face-first with his palm burning and some zetta weird puzzle on his phone about _missions_.

Fortunately, Sho is a quick study. Learns a few essential things:

  * _Whereas, Sho Minamimoto is now **dead** ;_
  * _Whereas_ , Sho Minamimoto is not only dead, but apparently is in hell of a sort;
  * _Whereas_ , he has seven days (168 hours, 10080 minutes, 604800 seconds, 1.118897906711 x 1049 Planck units of time) to actually solve these puzzles if he wants to exist in any form,


  * _Whereas_ , he is now functionally in a bounding box of approximately a radius of 2.193094 kilometers, which has as its limit the boundaries of Shibuya ward;


  * _Whereas_ , being in hell (or at least an alternate brane which involves at least SOME interaction with the familiar 3+1 oriented universe of the Shibuya he was familiar with when actually alive), there are limited points of crossover with what the dead (amusingly, to Sho) call the “RealGround”;


  * _Whereas_ , both professional griefers (known as “Reapers”, he’d find) and wave functions made of what are (quite accurately) described as “Noise” exist, and the latter will fuck up the dead such that they end up effectively “dead-squared” and themselves reduced to background energy;


  * _Whereas_ , there is at least a theoretical chance of escaping this gauntlet of fuck by playing their puzzle games-with-hazing;
  * ∴ One should play the _zetta factoring fuck_ out of this if he wants to get out of this in any sense.



And play it, Sho does. Possibly _too_ well, in fact; rather like the trig proofs (which was actually one of the bits of math in high school he occasionally would get dinged on) where the teacher would yell at him for “skipping logical steps” that he thought were inherent and screamingly obvious, Sho goes for the throat--and focuses on that end goal of _living through seven days of this even if my partner is a complete zero of a son of a digit who probably needed help blowing his nose in life._

At the end of seven days, Sho is alive. His partner is very much _not._ His first partner had been left to the Noise somewhere around day four, in what had actually been a fake-out bid to save his own skin. 

Needless to say, he is _not_ amused to be told he has Lost The Game...until he is offered the consolation prize of becoming a redshirt Reaper, as he’s already seen that the Reapers have more freedom…

Sho doesn’t even bother to ask what the price of entry for his soul was in the Game. 

The cost ends up being very obvious over a function of time.

* * *

Sho does not stay a redshirt Reaper for long. If he was a quick study at the Player’s Game before (well, other than the _don’t let your partner be killed_ Important Life Lesson), he’s a quicker one yet as a Reaper, and quickly learns the joys of being a complex number on the complex material plane...at least for a little while, anyways.

Megumi Kitaniji watches, is actually genuinely _impressed_ with the little cocky shit, as time goes on it’s obvious Sho has honest TALENT, from finding new ways of griefing the Players (he was amused when one day Sho had the Players actually attempt to calculate their coordinates in the complex plane…) to actually getting the hang of his _own_ energetic signature enough to get the wings under control.

At some point, the kid gets notified he’s earned his Reaper wings in the form of a job promotion to Game Master.

And at some point he manages to get _six_ of those wings, although nobody is really sure how. Not even Sho himself.

* * *

Even as Sho is finding his place in the UG, the price of his entrance is beginning to make itself known. 

The occasional math puns he made in life slip into his speech regularly, almost like a tic, a compulsion, an odd verbal stim. At some point, he’s honestly not aware half his slang is mathematical jargon; it doesn’t help that he’s actually found that the jargon actually has Power here, even more so than in the RG.

The habit he had of occasionally finding objects tossed on a street or in the trash that he’d turn into “modern art” has turned into an odd sort of hoarding, not unlike that of a magpie feathering its nest with various trinkets and toys. Eventually, the hoards turn into towers. The urge to Create, to align things just _so_ to create a symmetry that only Sho can see or comprehend, occasionally is overpowering.

Sho, before his death, could have listened to a lecturer on Youtube for hours. Now, Megumi Kitaniji is lucky if Sho can manage to stay in a meeting for fifteen minutes before he fucks off for “Arts and Crafts Time”.

And even as Sho had _pi_ as his favourite transcendental number in life, he has almost a religious reverence for it now. There was something _special_ about it, if he could only put his finger on it…

And then there is the incident with Sho when Yodai Higashizawa got promoted up the ranks. Kitaniji had to have a minor sit-down with Sho at that point and explain the basic concepts of “Please don’t call the new Game Master for week 1 ‘Ram-crotch’ in casual conversation, _especially when we’re having a meeting on planning the next Game_ ”. 

Anyone who’d known him in life would have known Sho Minamimoto had lost the plot and pretty much any concept of an internal filter, but...well...equations are _always_ balanced in some way; it’s a law of mathematics.

And if that balancing of the equation of _living as a literal shinigami_ got balanced out by _other things..._ well, so be it.

* * *

Over a function of time, Sho realises _why_ the Reaper gig is in fact the _consolation prize_ for losing the Game.

Yes, there was a bit more freedom in the Z axis, sure, but still limited points, and he’s still in that damn bounding box of Shibuya, _and_ at least some portion of his time that he could be using in Productively Fucking Off by making the throne of this garbage heap of a town has to be used to put out missions and sit in zetta boring _meetings_.

And if just ONE of these factoring hectopascals could just solve a simple Z-class problem it might actually make his day less _boring._

Sho desperately wonders if there’s a way out of the bounding box. Obviously Kitanji isn’t going to be leaving soon...is this all there is? Trolling dead kids all fucking day just to keep alive?

So it is quite a surprise one day when Sho finds one of his Trash-henge stelae of detritus tagged with Euler’s identity, and a note stuck in a vending machine’s coin return with the message: 

_Meet me at WildKat at time T._ _t=4/sin(23.4)_

_We have a shared interest in perturbation of the system._

Sho smiles and descends from the pile. His life just got _interesting_ indeed.


	4. hyper3: EXPONENTATION

When Sho shows up at the WildKat Cafe at precisely 10pm sharp, Sanae Hanekoma first offers him a coffee, then leads him into a back room that he had not noticed until then.

And that night is the beginning of an new axiom of choice, as Sho Minamimoto is let in on a whole series of variables and a whole new world of truths expressed in equations.

For the first time in his life, Sho has a teacher that is not only his equal, but his superior, and who actually could see the _beauty_ in the numbers, the equations...and who encouraged them. He sees it itself as a beautifully balanced equation, and he has needed this like a desert plant needs water.

Hanekoma, to his credit, is more than willing to oblige.

* * *

Sho learns about many, _many_ things in the following months.

He learns about Gödel’s set theory, and the axiom of choice, and transfinite sets. At one point he gets the idea that the RG and UG can actually be interpreted as different transfinite numbers and is even more fascinated when Sanae Hanekoma casually notes Gödel was actually doing this as a kind of a mathematical proof of God’s existence, of a higher plane of reality.

(Hanekoma mercifully does not tell Sho that Kurt Gödel starved himself to death after his wife passed on as he was utterly convinced people were out to kill him for his mathematical knowledge of the divine.)

He learns about calculus, and Isaac Newton, and how--after Newton and Leibnitz co-invented calculus--that Newton would become an alchemist trying to essentially perform a derivation and integration on the very nature of reality and the soul itself.

He learns about _E8_ and its intrinsic relation to many other Lie algebras, and how it is intrinsically related to Theories of Everything.

He learns the greater truth of how Noise--and this brane--operate. Not just the practical--pretty much upon Reaperhood and becoming a redshirt, one is taught how to call forth the Noise--but the _theoretical_ , such as the Noise being an expression of a function in a sea of imaginary numbers which intersects with the real line forming the complexity of the UG.

He learns Hanekoma has a hobby. A very _interesting_ side hobby, as the most famous street artist in Shibuya. And the whole point of the exercise--as Hanekoma, no, CAT reveals--is to allow people their own axiom of choice.

He learns that Sanae Hanekoma groks him better than possibly any other individual, living or dead, EVER has. He discovers possibly the closest he’s ever known to what he’d term as a friend.

Sho even discovers he likes a proper macchiato--not the overtly sweet shit at the American cafe near the Scramble, but actual _macchiato_ , made with coffee black and bitter and the fuel of life itself.

And one day, in one of these lessons he learns of the existence of the Composer, the twink bastard actually rigging the whole Game...and that they are in agreement that a change is needed in management.

* * *

At one point, Sho dreams. Usually his dreams are of wheels within wheels, or of mathematical beauty, or the mundane sort of a small iteration of one’s day.

This one is different. His grandmother, who used to read him stories. Stories about things that never happened, but which comforted him anyways.

The dream he has is of his grandmother telling him the story of the Monkey King and how he managed to almost overthrow the gods.

Sho wakes up on one of his trash-henges, and smiles. Just a bit more, a bit longer, and everything would change.

And just like Monkey, nobody was gonna put him in a goddamn cage or give him a polished-turd title again.

* * *

At one point, Sho finds another little package stuck in his trash-heap representations of the world as garbage and vanity (and also, he mused, of a program’s memory space; really, the garbage of the world WAS its memory cast aside and forgotten).

It’s a gun, and a message--the Composer can be taken out in the RG, and is planning something rash.

He goes--tunes, vibes--down, to the brane below, where he witnesses Joshua with a gun. (This is _definitely_ not in the game theory of how people are meant to end up playing the Game; Sho knows _that_ much.)

He fires. Is completely gobsmacked to find the bullets stopping just short, like that factoring twink was goddamn _Neo_ or some shit...just before HE’S shot in the arm, and then the kid with the headphones gets shot and bleeds his life out in front of the CAT mural.

It is at this point Sho Minamimoto comes to the conclusion that Joshua is, in fact, a Fractal Bastard; an actually _recursive_ Bastard whose Being A Bastard is evident at every scale of resolution in self-similarity of Bastardy.

* * *

When Sho stumbles into the WildKat Cafe holding a sore right arm, Hanekoma already knows what has happened, but asks anyway. His student--his future _Conductor_ , no, _Composer_ \--needs to get it out.

Sho proceeds to quite angrily rant on about _that fractal bastard_ and _that yoctogram just got shot like nothing and this is the binomial who’s running the Game?_ and _what the factoring fuck kind of function lets you stop bullets?_

Hanekoma bandages him up, and reassures Sho that there are multiple ways to calculate pi in the end, and begins the next phase of Sho’s training.

One of the first lessons being that, much as with quantum physics and observation, the nature of Noise and calling it forth is in fact not only sensitive to initial conditions but observation. All things are wave functions in the end, after all.

Hanekoma knows there isn’t much time now. He knows all too well what the Composer has planned.

Hanekoma knows that literally everything he has worked for in tending this small garden is at threat.

The lessons shift after this. Sho has been introduced to the concept of calculus and esoterica and the nature of the Noise (in that it is functionally a boundary condition, like the edges of a Newton fractal); now there is practice for what is surely to come.

* * *

A week after that, the Fractal Bastard shows up in the Game--as a _player_ , at that!--and Sho enters the back room of the WildKat Cafe where the ever-changing library (which somehow always has the appropriate study material at hand) is, a veritable storm of bifurcations in his speech ranging between feral laughter on how he’s finally going to _get_ that factoring binomial and ranting on wondering what the hell the Composer was planning.

(The rant is almost, but not quite, a sine wave of emotion between the positive and negative.)

Hanekoma blinks. This was unexpected. This...complicated things, and it was time to pull out the failsafe.

* * *

The lesson plans change, and over the next week--incorporating an idea _Sho_ comes up with involving the _Euler identity_ , mixed with some encoding in basic Enochian and the literal language of angels--Hanekoma gives Sho the final lesson that will be the damnation of them both.

The final lesson is the nature of untamed Noise, and its summoning and refinement, without the normal bounds of behaviour.

Such a thing is hacking the code of reality, and like other forms of hacking, highly illegal. Immoral.

_Taboo._

* * *

Even as his work with Hanekoma progresses on what is effectively divine treason, Sho’s control--what little he has left, anyways--slips further.

He pretty much gives minimal effort to the usual harassment. The Game is zetta dull, like his courses in high school designed to feed pablum when he desired meat. What he’s doing with Hanekoma, on the other hand...this is _relevant to his interests_. There is Truth and Beauty in what he’s doing, and he finds a higher value in his experiments in the back room of the cafe than he does in what he increasingly regards as busywork.

By the time that the Composer has entered the game (which is technically not quite within that axiomatic set of rules, but since when did rules matter? Rules are garbage; he’ll deal away with that soon enough) there are days he forgets to send missions altogether.

By this time, too, the griefing that Reapers do to the Players in the game has taken on a definite _edge_ \--when Sho can be bothered to really focus on it, that is.

What he’s doing now is _far_ more interesting. He’s reverse-engineering his own code, and that of Shibuya, and when the God Factor is removed from the equation...he’s going to rewrite this function in his _own_ image.

* * *

On the fourth day of the second week, Sho decides to play a prank on the guy who sneered at his math-rock fascination in life. (He had his own band now, of course.)

Sees (of all things) one of the members of Def Märch leaving the band’s mike (the one they’ve had for fucking forever) in the Phone Booth of Love. 

Sho switches this out with the old megaphone HE’s had since forever, and promptly puts the old mike in the Memory Heap (as Sho has taken to affectionately calling his trash-henge to himself).

Before the switcheroo happens, he makes sure to record himself screaming about sines and cosines and tangents into the mike. 

Sho won’t need the megaphone for much longer anyways. But before setting his plan into motion, he wants those radians to know that his voice is going to shake the _heart._

* * *

Shortly after he’s done his farewell serenade, Sho sets up the variables for this calculation that (at least he himself) sees as flawless.

Finds a back alley in Udagawa. Far enough that it won’t be noticed straightaway. He knows his own variable is to be removed if this particular formula is discovered.

Attunes himself, centers himself to the zero point of (0,0) for the graph, for drawing what is intensely personal calculus in a language of the divine, for the purpose of hacking reality.

For defeating Erasure.

For allowing himself to be integrated after he expresses the derivative of himself in the function of wiping out that tasteless tetrahedra of a Fractal Bastard.

The sigil itself incorporates _pi._ He recites the formula to himself as he scribbles in Enochian and the universal language of numbers in chalk and Vibes, with it, _becomes_ it in a sense. It is math, and mnemonics, and magick all in one..

_Three is the point of the one._

_Four to the one-five-nine are two._

Numbers are alchemy are Art.

 _Six-five,_ as he draws while facing to the North, the positive Y axis.

 _Three-five,_ as he draws while facing to the West, the negative X axis.

 _Eight-nine,_ as he draws while facing to the South, the negative Y axis.

 _Seven-nine_ , as he draws while facing to the East, the positive X axis.

 _Three two three eight four_ as he draws the marking for the Mind.

 _Six two six four three_ as he draws the marking for the Body.

 _Three eight three two seven_ as he draws the marking for the Spirit.

The mind and the body and the spirit form an isoceles triangle solving for the azoth of the philosophers in the complex plane solving for the arithmetic progression with a common difference of 4. The four directions serve as that which defines the real and the imaginary in the complex plane in the X and Y axes.

_Solve et coagula._

In his excitement and haste as he sets his doom in motion before he is discovered, he does not notice he does the equivalent of forgetting to carry over a sign.

* * *

The Composer ( _that fractal bastard_ ) himself shows up with the emo kid on the final day, and Sho knows exactly what is to be done. He’s set up this equation, anyways.

There is truth in numbers, and this world is a complex function.

At the centre of so many functions, a sort of root of reality itself (so Sho has determined) is _pi._

The great secret of the UG is that the imaginary is the driving axis as the real line is in the RG.

Sho laughs--he wasn’t quite expecting them this early--but like his old trig proofs, that means he can skip a step. A happy miscalculation.

He takes the leap of faith that--upon what he is about to do--his own derivative will be integrated via the taboo noise sigil pointing to [](https://www.codecogs.com/eqnedit.php?latex=%5Cint_%7B0%7D%5E%7B%5Cinfty%7Df\(x\)dx=%5Csum_%7Bn=1%7D%5E%7B%5Cinfty%7D%5Cfrac%7B1%7D%7Bn%5E8%7D) with himself as the variable.

He cackles. 

“3.141592653589793238462643383379502884197169399…”

Letting the very truth of the universe flow through him, or at least the closest thing to truth. He’s gonna be out of this goddamn bound soon.

They _all_ are. It's just a function of time now.

“3751058209749445923078164062862089986280348253421170679821480865132…”

Everything’s going to be fixed soon. He’ll write a new equation when he comes back. One with no goddamn leashes or bounding boxes. One that is infinite. _Transfinite._

“8230664709384460955058223172535940812848111!”

One of the truths of the universe is that _e iπ+1=0_.

He will use the very power of pi to logarithmically summon forth the imaginary to zero these out.

Everything is flawlessly calculated and, in Sho’s mind, _indescribably_ beautiful.

As the level _i_ flare goes off, Sho Minamimoto thinks...not of math for once, but a song he used to hear in the record store before his death:

_If man is five then the devil is six_

_The devil is six, the devil is six_

_The devil is six, and if the devil is six_

_Then god is seven, then God is seven_

_...this monkey’s gone to heaven_


	5. hyper4: TETRATION

Sho--at least the tattered remnants of that bit of soul still having enough of a self-concept to see himself as Sho--finds himself in a not-light that is not-darkness and an infinite number of souls such that all is drowned to white noise. Static.

He knows to stay here is to risk ego loss. Searches for the anchor desperately as an infinite number of piranhas gnaws on his soul.

Sometime on the morning of the third day, Sanae Hanekoma finds the sigil, finds the dropped jot, the slightly askew tittle, the small things that would allow this to work were it not for these entirely too human errors.

And--as much as time actually existed in a place which was a border domain, like the boundary region of a Newton fractal--the link is found, and eventually Sho finds himself pulled back to the UG, just as surely as _e iπ +1=0 _.

* * *

The taboo noise sigil, he remembers from Hanekoma’s discussions, is effectively a number sieve of sorts; its purpose being much like a calculus equation, to find the derivative of a known soul in the Noise and integrate that with the physical.

Whilst Sho didn’t come back entirely the same, he came back _right._

There were certainly...side effects, some deviations, but these could be worked around; the first thing to work on now was finding that goddamn prissy _motherfuck_ of a factoring hectopascal that dared call himself _God_ that somehow bamfed out at the last fucking attosecond.

He will fucking _break_ out of this sandbox that is Shibuya and the UG, even if he has to drop the fucking _moon_ to do it.

* * *

It soon becomes obvious that...Sho had in fact both come back Wrong, and possibly all _too_ Right, and both at the same time.

His behaviour becomes even more erratic. The piles of garbage even less in balance than before. 

In a blind rage, he ransacks the WildKat Cafe, looking for Hanekoma, for that kid with the headphones that was with Joshua, for Joshua himself, ANYTHING that will get him closer to doing a full code rewrite such that he can just...balance...this fucking _equation_ , rid the bounding box, get rid of that one annoying variable.

He ransacks the WildKat Cafe not just looking for Joshua or Hanekoma, but because he has come to his own terrible conclusion.

He is in a function with sensitive dependence on initial conditions, and he is in a place where the slightest nudge either way will throw his eigenstate towards either zero or infinity.

And honestly...the math he has done is not showing a trend towards the _infinity_ option. The tabooing-up...it’s spreading. Getting worse. Actually making it harder for him to do the calculations he needs to do--it is like a weird spiritual shingles, a spiritual _rabies_ , a red-hot poker pressing in the back of his mind and his soul even as the tattoos burn in a thousand kinds of pain...when he can feel those portions of himself at all.

He has to solve this equation. Has to escape the bounding box. Has to write his own axiomatic theory, because he is completely and totally _zetta fractally fucked_ _at every resolution_ if he does not.

* * *

He eventually finds Joshua near one of his trash-henges, and as he rushes in to attack--threatening to have the taboo take him over entirely--the Composer collapses the wave function--and the trash--upon Sho.

Sho has not had time to redraw the sigil--probably cannot concentrate at this point with the white noise and the weird fucking _chanting_ at this rate vibrating the UG--and knows this function has reached its endpoint.


	6. hyper5: PENTATION

Again, there is not-darkness and not-light, and the white noise of a sea of millions--billions?--of souls in a sea. Again, Sho desperately tries to keep his head afloat metaphorically in a sea with no top or bottom and which is every bit as non-orientable as a Klein bottle.

For a while--a brief, brief while, in what is his last vestiges of consciousness--Sho actually has a thought that his situation is like a story he heard from his grandmother when he was very young, when the king of the monkeys got trapped under a hill for 500 years for trying to overthrow Heaven and nearly succeeding.

He laughs bitterly, and the complex component of Sho Minamimoto goes entirely too still, and the imaginary component of Sho Minamimoto desperately tries to keep from drowning in a sea of infinite numbers.

* * *

Sometime later...after days? Months? Years? (Did time have _meaning_ , when an area has been in a closed timelike curve weaved out of Noise?)...a fairy-kei nightmare giggles as she creates a Taboo Noise sigil. 

The young child (who is very much _not_ a child) knows _the veil itself is thinning,_ and she must act quickly.

The young child (who is very much _not_ a child) going by the name of Coco Atarashi has _plans._ _EPIC ones._

And these plans require a certain Neku to make one more trip through the UG, and...well...she knows just the _omfg perfect zetta strong partner_ for the job.

And so Sho Minamimoto finds himself pulled again from the plane of the imaginary into that of the complex by an external variable that certainly put the _faerie_ into “fairy kei” aesthetic.


	7. Appendix A: Terminology and references

Yes, I am an incorrigible Math Dork, and there are quite a lot of mathematical references and in-jokes and such that may not be immediately obvious--but are important to note. Also, there are some _non-maths_ refs in this as well worth noting, and which are a little less “story jarring” to put explanations of below.

The title “ _z ↔ z² + ci_ ” is itself the classical expression to derive the [ Mandelbrot set](https://www.medicographia.com/2013/01/the-maverick-mathematician-benoit-mandelbrot-and-the-stunning-beauty-of-the-fractal-universe/), which is (mathematically speaking) considered to be a set which (via transformation) represents all Julia sets (and, strictly speaking, the formula is the initial expression showing the “main” set). It is an infinitely complex, internally-connected set derived literally from squaring a complex number and adding a (second) complex number representing X and Y coordinates. If the resulting function (iterated) trends to 0, it’s within the Mandelbrot set; if it expands to infinity, it is outside. (Most programs that do generation of the Mandelbrot set use a check of “does it appear Z is becoming larger than 2 or -2 on real or complex scales?” as a rule.)

The Mandelbrot set also, as it turns out, [ has some surprising connections to mathematical constants and formulae](https://en.wikipedia.org/wiki/Mandelbrot_set#Pi_in_the_Mandelbrot_set). In 1991, it was found that for a specific value of z (and working backwards), the Mandelbrot set can actually be used to approximate pi; the Mandelbrot set also has some fundamental connections to the Fibonacci sequence, specifically in how “threads” split off from the main sequence.

It is (in my opinion) one of the most beautiful objects in mathematics, and is so simple to calculate and graph using modern computer equipment, and there is a deep and internal truth and beauty to it that still brings me to happy tears to this day; there’s a reason I have [ a wall poster ](https://www.mandelmap.com/) of a mapping of “interesting” bits of the Mandelbrot set along the boundary line.

The “hyperN” list refers to specific orders of [ hyperoperations](https://en.wikipedia.org/wiki/Hyperoperation), aka “levels of mathematical operations”. Two that are probably not that familiar are [ tetration ](https://en.wikipedia.org/wiki/Tetration) and [ pentation](https://en.wikipedia.org/wiki/Pentation); tetration can be _roughly_ described as “fun with making an exponent, and then pretty much putting that exponent in itself in a heap of a ‘power tower’”, and pentation is essentially making heaps of tetration towers. This is being used in a stylistic sense, showing how, well, Sho iterating himself in a sense (and the resulting function) would be seen--assuming you asked him about it.

Tool’s “[The Holy Gift](https://unbumf.com/tool-fibonacci-secret-album-holy-gift/)” (for the non-Tool fan) is an alternate (and one could argue, probably the _original_ intended arrangement) of _Lateralus_ which fits the arrangement of the tracks in such a way that all lyrics and time signatures on the album actually spiral down in the [ Fibonacci sequence ](https://en.wikipedia.org/wiki/Fibonacci_number) , which (when divided by descending members of itself) can be used to approximate the [ Golden Ratio](https://en.wikipedia.org/wiki/Golden_ratio); when the numbers of the Fibonacci Sequence are plotted (via squares) a spiral is formed--a spiral which actually occurs remarkably commonly in nature, [ especially with flowers](https://en.wikipedia.org/wiki/Fibonacci_number#Nature). There are also interesting relationships with prime numbers, hypotenuses of right triangles (starting with the number 5 in the sequence), and so on. The title track of _Lateralus_ even contains clues (of the sort mathgeeks would be all over!) on how to do the “Holy GIft” arrangement.

(Yes, you’re reading this right: Tool made a veritable _extended Minamimoto-esque mathrock arrangement that is probably actually the original arrangement._ ) This is pretty much the sort of thing Sho would just find to be the Zetta Coolest Thing EVER, and I know _I_ certainly did on finding this out!)

[ Voivod ](https://en.wikipedia.org/wiki/Voivod_\(band\)) is...a wonderfully eccentric Quebecois metal band that originally started out as speed metal and has evolved (over the years) to effectively something I’d term “progressive math thrash with elements of alternative” (it’s actually pretty difficult to describe Voivod’s _actual_ musical genre starting with _Dimension Hatröss_ ). I would argue that, in my teens, they were the band that introduced me to the concept of “math rock”--especially as _Nothingface_ includes “Into My Hypercube”, which is _literally_ from the viewpoint of someone in a tesseract (although it’s also...much weirder than that). 

[ _Karoshi_](https://en.wikipedia.org/wiki/Karoshi), aka death by overwork, is an actual thing--especially as there is a strong societal expectation that one dedicates one’s life to one’s workplace. (That said, there are starting to be workplace laws in Japan cracking down on practices that cause employees to literally work themselves to death, including a worker’s comp program for people dying of _karoshi_ .) The concept of a “ [ black company](https://en.wikipedia.org/wiki/Black_company_\(Japanese_term\))” is also related--it’s a Japanese term for a corporate sweatshop of a particularly abusive nature. (The office in “Aggretsuko” is essentially a “black company”; the term even shows up in _Cells At Work: BLACK_ (released as _Cells At Work! Code BLACK_ in English), which functionally portrays cells living in someone with Very Bad Health Habits (and who is also dying of karoshi to boot!) as working in the equivalent of a “black company”.)

Yes, mathzines actually do exist in Japan, and the particular “math doujinshi circle” mentioned is a real one--[ The Ankokudan ](http://ankokudan.org/d/d.htm?index-E.html) aka the [ Darkside Communication Group](https://en.wikipedia.org/wiki/Darkside_communication_group). And yes, they actually do show at Comiket, and (among other things) publish two magazines ( _Monthly Pi_ and _Monthly Zeta_ , which are actually [ publications of digits of pi](https://soranews24.com/2015/06/08/mysterious-japanese-publishing-group-releases-book-with-pi-to-one-million-places/) and the imaginary values of the Riemann zeta function). The group is also a bit of an avant-garde (and esoteric) art group when they’re not doing Math Books As Art, and even have an actual discount at their table based on how many digits of pi one has memorized--it’s something all in all that’s _zetta fucking cool_ and something Sho would TOTALLY vibe with.

The [ mathematical universe hypothesis](https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) actually does exist (and honestly is part of effectively a “multiverse” math model, yes, branching timelines and the [ many-worlds interpretation of quantum mechanics](https://en.wikipedia.org/wiki/Many-worlds_interpretation) also fit in)--and (just as it says on the tin) pretty much postulates all of reality is definable in mathematical functions. Arguably it’s a subset of [ mathematicism](https://en.wikipedia.org/wiki/Mathematicism).

The [ Newton fractal](https://en.wikipedia.org/wiki/Newton_fractal) is another fractal set (one that is, if anything, a “Mandelbrot cousin” as it’s actually a specific Julia set generated by a [ specific formula for finding roots named after Isaac Newton](https://en.wikipedia.org/wiki/Newton%27s_method)) that has some extremely interesting activity along boundary regions (where it’s almost, but not quite, escaping the function). It’s also a very attractive fractal, and one I chose for both the appearance of its boundary regions (which reminds me a lot of the description of the nature of Noise from the Secret Reports) and it (and its underlying root-derivation function) being named for Isaac Newton--who not only was a co-inventor of calculus, but dabbled quite a bit in esoterica.

The references to [ Enochian](https://en.wikipedia.org/wiki/Enochian) are from an esoteric alphabet (and associated esoteric language) that was initially written down in journals of John Dee and Edward Kelley (Dee was a ceremonial magick practitioner and alchemist, and Kelley was a spirit medium who assisted Dee) as part of a generalized magickal system Dee created in the Elizabethan era. (It would be rather appropriate for an angel instructing a Reaper, at any rate.)

Some of the details on math being used to encode sigils was in part thanks to discussions with a beta reader (Willow) as well as some more direct inspiration from things like isopsephy (the Greek and Aramaic practice of “number punning”--their “Roman numerals” equivalents covered the entire alphabet--that gave rise to gematria in Qabalah and in modern ceremonial magick systems). Honestly I really do think Sho explicitly used _pi_ as a magickal formula of sorts, and specifically in the sense of “spiritual alchemy” systems, hence the inspiration. (There is also some minor “four is death” symbolism, which actually worked out REALLY elegantly figuring out where the hell “159” would fit in some weird math ritual based on [ arithmetic sequences](https://primes.utm.edu/glossary/page.php?sort=ArithmeticSequence).)

The [ hermetic imagery](http://www.esotericonline.net/m/blogpost?id=3204576%3ABlogPost%3A957084)...pretty much is something that lent itself, due to the links of early mathematicians (up to at least the Enlightenment period, really, into the modern era if you count Gödel) and esotericism, and some...very interesting balancing of equations that happened with the interactions of the game (and particularly the Secret Reports), that history, and at least some writings on alchemical imagery. (In particular, the specific imagery of Valentinus on the process of alchemy--and hermeticism--being a process of self-refraction, and the specific geometric imagery of his particular Azoth Mandala, comes to mind.)

One of the formulae noted is a (bastardized) form of the [ Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function), which is (admittedly) being used pretty much as nonce here as part of what amounts to an extended mathematical pun based on Sho’s nickname among the Japanese fandom. 

(And if anyone had told me I was going to need to go back to my college years of LaTeX formatting for mathematical functions in a fanfic even five years ago, I’d have told them they were mad. :D)

The specific use of _pi_ in the Taboo noise refinery sigil...turns out to be _very_ happily quite canonical! It shows up in a screen where Hanekoma is “dealing with”--in truth, fixing--said Sigil, and there’s actually [ a wonderful screenshot here that shows it in most of its Tabooty glory](https://vignette.wikia.nocookie.net/twewy/images/0/00/Taboo_sigil.png/revision/latest?cb=20190424040654) (and [in even better definition in an "RG install" at the Square Enix ARTNIA Cafe in 2018](https://www.famitsu.com/images/000/166/994/z_5bdc16118a6da.jpg)). Not only does it actually include _pi_ to the 29 decimal places noted, but includes some other interesting references (definitely to the natural logarithm _e_ and to the Golden Ratio, a possible reference to _i_ , and even a (non-mathematical) tagging of the sigil in Romaji (as M. Sho) with the “o” having little ears and a tail; there are also references to the (Western) zodiac in an ordering closer to modern Gregorian calendars, and even some interesting bits where the positions in the Zodiac are written out in English (with the actual numbers incorporated!) and looking _very_ Enochian-esque if looked at from certain angles. (It’s the little details like this that show me the love that was poured into this game, and I’m at least glad to see a personal headcanon confirmed even as I was writing this fic.)

As has been probably guessed, there's at least some inspiration from the song "[Transformation](https://twewy.fandom.com/wiki/Transformation_\(track\))" in the TWEWY soundtrack; I had _fun_ with that, and if anything was the original root function of this work, it was probably this. The title is also more directly a pun relating to [mathematical transformations](https://en.wikipedia.org/wiki/Transformation_\(function\)), as well as the general transformations in worldview we have as we grow and develop.

I...also couldn’t resist some other minor bits of punning for reasons of thematics, specifically on certain kanji readings of Sho’s name and at least some parallels I personally saw with, well, Monkey and his story. (Right down to being buried under a mountain and then freed from imprisonment as a bodyguard to someone doing a Very Important Mission.) Also, it gives me an excuse to throw the Pixies into the mix. :D


	8. Appendix B: Author’s Notes

**Summary for the Chapter:**

> Please forgive the author for this garbage pile of a fic, thank you.

Thanks first off to all the folks at Square Enix involved with the creation of _The World Ends With You_ ; to a certain group of folks who got me into this madness (and You Know Who You Are if you ever see this!); to Willow, Luka, and Jaynis on Twitter and Discord for bouncing ideas back and forth re Hanekoma/Sho interactions, headcanon-y stuff, and initial beta reading; and last--but not least-- _you_ for reading this lovely garbage pile of a fan drabble that somehow expanded exponentially into an off-season NaNoWriMo project. :3

* * *

In a way, this particular fic is itself a bit of a derivative function.

I was originally introduced to _The World Ends With You_ back in 2010-2011ish by some good friends who had picked it up, played it through, and informed me that I needed to get the game because...well...they felt Sho reminded them of me. (NO COMMENTS FROM THE PEANUT GALLERY JUST BECAUSE I OWN A MANDELMAP. :>3)

And oh dear gods, as it turns out...I find Sho just so...goddamn... _relatable_ , both on the “probably autistic and definitely feral and definitely having a _very unhealthy interest_ in mathgeekerie” and other aspects, the whole “yes, I am not and will never be Tame, I will pretty much ramp up my efforts to at least a logarithmic level of growth in pursuit of a goal especially if someone tells me it’s Impossible, occasionally to insane or unhealthy levels (yes, including pushing myself to goals where sometimes mistakes happen, pushing myself towards a goal such that I have to be explicitly reminded to Step. _Back_.) Oh, and wanting to punch the basic order in the face in general. >:3

So it was played, loved, put on the metaphorical heap when no sequel was incoming (I have had enough experience with Eternally Suspended Franchises to not get my hopes up too much--hello Breath of Fire cough ahem)...only to see _Final Remix_ released in 2018 (with new content!) and--most recently--the TWEWY anime announcement for 2021. (At least now I can show my friends who aren’t _gamers_ at least SOME of what I love about it, even if the anime ends up being the functional equivalent of a three-dimensional shadow of the four-dimensional hypercube of the original source material.)

And pretty much, just like a randomness seed, in its own bit of sensitive dependence on initial conditions...well.. _.this_ fic happened. (And other related fics too, which are hanging off this like bifurcation branches off a fractal, but we’ll get to those in a function of time. Right now, the root function needs to be established.)

And in the process of writing this, well...it has been a while since I’ve had THIS persistent of a muse in my head, to the point it’s honestly the first time I’ve done fanfic at _all_ since drabbles on Livejournal literally over a decade ago at this point. (Honestly, even this fic has become part of an entire series as I’ve been in the process of writing and storyboarding ideas in my head--it’s certainly gotten me out of my own comfort zone and pushed me to expand my creative world, just a bit.)

In discussions I’ve had with my beta reader on this, they mentioned TWEWY is a transformative work--not just in terms of storytelling, mind, but in the lessons and ideas we pick up on it. Much like a bifurcation diagram, there is some sensitive dependence on initial conditions, but...we always get out of it what’s needed at the time; the math does go to the solution, though occasionally--like Benoit Mandelbrot’s happy accident in 1980 in the IBM labs in an attempt to map the simplest transformation of Julia set coordinates that the title refers to--the solution is not always the expected or anticipated one.

...Yeah, maybe the friends did have a point. :D But, well, if this particular fic does serve as the derivative function to maybe give a glimpse in the beauty (and, occasionally, the...not-beauty) Sho sees in the world, lets one see that map on the complex plane of what it’s like to be in his head...I’ll say I’ve definitely arrived at my _desired_ solution.

Q.E.D. Class is dismissed.


End file.
